Finite Group Databasebeta

← #29669585

Subgroups — #29669585

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458800C211yesmaximal, characteristic, complemented
2#21458749C211yesmaximal, characteristic, complemented
3#21458486C211yesmaximal, characteristic, complemented
4#21445445C211yesmaximal, characteristic, complemented
5#21445300C211yesmaximal, characteristic, complemented
6#21445233C211yesmaximal, characteristic, complemented
7#21445194C211yesmaximal, characteristic, complemented
8#21445165C211yesmaximal, characteristic, complemented
9#21445019C211yesmaximal, characteristic, complemented
10#21444946C211yesmaximal, characteristic, complemented
11#21444914C211yesmaximal, characteristic, complemented
12#21444896C211yesmaximal, characteristic, complemented
13#21444655C211yesmaximal, characteristic, complemented
14#21444638C211yesmaximal, characteristic, complemented
15C61×#3287C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840491C2211yescharacteristic
18#16840170—21no—
19#16840160C2211yescharacteristic
20#16840134—21no—
21#16840134C2211yescharacteristic
22#16840134C2211yescharacteristic
23#16840108—21no—
24#16840108C2211yescharacteristic
25#16840108C2211yescharacteristic
26#16840067—21no—
27#16840057C2211yescharacteristic
28#16840052—21no—
29#16840052C2211yescharacteristic
30#16840047C2211yescharacteristic
31#16840045—21no—
32#16839967—21no—
33#16839967C2211yescharacteristic
34#16839959—21no—
35#16839959—21no—
36#16839959C2211yescharacteristic
37#16839959C2211yescharacteristic
38#16839892C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41#16839869—21no—
42C4×#11758200—21no—
43C4×#11758200—21no—
44C4×#11758200C2211yescharacteristic
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839507C2211yescharacteristic
49#16839506C2211yescharacteristic
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56C2×#135×C61—21no—
57C61×#550C2211yescharacteristic
58C61×#550C2211yescharacteristic
59C61×#493C2211yescharacteristic
60C61×#466—21no—
61C61×#460C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#449—21no—
64C61×#449—21no—
65C4×Q16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200—41no—
73C2×#11758200D411yescharacteristic
74#14519777—21no—
75#14519777—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—21no—
79#14519776—41no—
80#14519768—21no—
81#14519768—41no—
82Q8×Dic61—21no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×Dic122C2311yescharacteristic
87#14519710—21no—
88#14519709—21no—
89#14519709—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
102#14519700D411yescharacteristic
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110#14519687—21no—
111C4×#92418—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C2×#41×C61—21no—
117#135×C61—44no—
118C2×Q16×C61D411yescharacteristic
119C2×SD16×C61—21no—
120C2×SD16×C61—21no—
121C2×D8×C61—21no—
122C2×D8×C61—21no—
123C22×C8×C61—21no—
124#127×C61—21no—
125#127×C61—21no—
126#127×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#121×C61—21no—
129#121×C61—21no—
130#121×C61—41no—
131#121×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61D411yescharacteristic
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—42no—
147#41×C61—81no—
148#41×C61—81no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61—41no—
160C2×D4×C61—41no—
161C2×D4×C61C2×D411yescharacteristic
162C22×C4×C61—21no—
163C22×C4×C61—41no—
164C22×C4×C61—41no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168C2×Dic122C2×D411yescharacteristic
169Q16×C61SD1614yescomplemented
170SD16×C61—42no—
171SD16×C61—42no—
172D8×C61—42no—
173D8×C61—81no—
174C2×C8×C61—21no—
175C2×C8×C61—42no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C2411yescharacteristic
189#11758200—41no—
190#11758200—42no—
191#11758200—42no—
192#11758200—81no—
193#11758200—81no—
194#11758200—81no—
195#11758200—82no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—41no—
201C4⋊Dic61—41no—
202C4⋊Dic61—41no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—21no—
205#11758196—41no—
206#11758196—41no—
207C4×Dic61—21no—
208C4×Dic61—41no—
209C4×Dic61—41no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216C23×C61—21no—
217C23×C61—41no—
218C23×C61—41no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61C2×SD1612yes—
222Q8×C61C2×SD1612yes—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—82no—
234D4×C61—82no—
235D4×C61—81no—
236D4×C61—81no—
237D4×C61—161no—
238D4×C61—161no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—82no—
247C2×C4×C61—82no—
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255Dic122—21no—
256Dic122—21no—
257Dic122—41no—
258Dic122—41no—
259Dic122—81no—
260Dic122—161no—
261C8×C61—21no—
262C8×C61—41no—
263C8×C61C2×SD1612yes—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#19770—611nomaximal, Sylow
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61D4211yescharacteristic
281C4×C61—22no—
282C4×C61—22no—
283C4×C61—21no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61C22×SD1611yescharacteristic
288C4×C61#57511yescharacteristic
289C4×C61#57711yescharacteristic
290C4×C61#57711yescharacteristic
291Dic61—41no—
292Dic61—41no—
293Dic61—81no—
294Dic61—81no—
295#3392—611no—
296#3387—611no—
297#3362—611no—
298#3287D6111yescharacteristic
299#1715—611no—
300#1688—611no—
301#1678—611no—
302#1671—611no—
303#1666—611no—
304#1642—611no—
305#1631—611no—
306#1626—611no—
307#1623—611no—
308#1582—611no—
309#1579—611no—
310C2×C61—81no—
311C2×C61—161no—
312C2×C61—161no—
313C2×C61#327111yescharacteristic
314C2×C61D4×SD1611yescharacteristic
315C2×C61#328811yescharacteristic
316C2×#135—21no—
317Q82—611no—
318#557—611no—
319#550C2×D6111yescharacteristic
320#550C2×D6111yescharacteristic
321#501—1221no—
322#498—611no—
323#493C2×D6111yescharacteristic
324#492—611no—
325#492—611no—
326#492—1221no—
327#486—611no—
328#486—611no—
329#486—1221no—
330#479—1221no—
331#477—611no—
332#476—611no—
333#476—1221no—
334#475—611no—
335#475—1221no—
336#466—21no—
337#461—611no—
338#461—1221no—
339#460—611no—
340#460—611no—
341#460—1221no—
342#460—1221no—
343#460C2×D6111yescharacteristic
344#460C2×D6111yescharacteristic
345#449—21no—
346#449—21no—
347#446—611no—
348C8×Q8—611no—
349C8×Q8—611no—
350#441—1221no—
351C4×Q16C2×D6111yescharacteristic
352C4×SD16—611no—
353C4×SD16—1221no—
354C4×SD16—1221no—
355C8×D4C2×D6111yescharacteristic
356#334—611no—
357#334—611no—
358#333—611no—
359#332—611no—
360#331—611no—
361#331—611no—
362Q8⋊C8—611no—
363Q8⋊C8—611no—
364Q8⋊C8—611no—
365D4⋊C8—611no—
366#322—611no—
367C61#1977011yescharacteristic, complemented, Sylow
368C2×#41—21no—
369C2×#41—21no—
370#135—44no—
371C2×Q16C22⋊D6111yescharacteristic
372C2×SD16—21no—
373C2×SD16—21no—
374C2×SD16—611no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—2441no—
381C2×D8—21no—
382C2×D8—21no—
383C22×C8—21no—
384C4⋊Q8—611no—
385C4⋊Q8—611no—
386C4⋊Q8—1221no—
387C4⋊Q8—1221no—
388C4⋊Q8—1221no—
389C4⋊Q8—2441no—
390#127—21no—
391#127—21no—
392#127C22×D6111yescharacteristic
393#127C22×D6111yescharacteristic
394#125—1221no—
395#125—2441no—
396#121—21no—
397#121—21no—
398#121—41no—
399#121—41no—
400C4×Q8—611no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8C22×D6111yescharacteristic
406C4×Q8C22×D6111yescharacteristic
407C4×D4—21no—
408C4×D4—21no—
409C4×D4C22×D6111yescharacteristic
410#107—1221no—
411#107C22⋊D6111yescharacteristic
412#106—611no—
413#106—1221no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—1221no—
421C4⋊C8—1221no—
422C4⋊C8—1221no—
423C4⋊C8C22×D6111yescharacteristic
424#103—611no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103C22⋊D6111yescharacteristic
429#103C22⋊D6111yescharacteristic
430#102—21no—
431#102—21no—
432#102—611no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—2441no—
439C22⋊C8—21no—
440#97—1221no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—1221no—
444C4×C8—1221no—
445C4×C8C22×D6111yescharacteristic
446#41—42no—
447#41—42no—
448#41—81no—
449#41—81no—
450C2×Q8—611no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—2441no—
454C2×Q8C2×C22⋊D6111yescharacteristic
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4D4×D6111yescharacteristic
467C22×C4—21no—
468C22×C4—41no—
469C22×C4—41no—
470Q16#1175820014yescomplemented
471SD16—42no—
472SD16—42no—
473SD16—4881no—
474SD16—4881no—
475SD16—4881no—
476SD16—4882no—
477SD16—2441no—
478SD16—2442no—
479SD16—2442no—
480D8—42no—
481D8—81no—
482C2×C8—21no—
483C2×C8—42no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8C2×C22⋊D6111yescharacteristic
491C2×C8C2×C22⋊D6111yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—2441no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4D4×D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C4⋊C4C2×C22⋊D6111yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C42—21no—
512C42—21no—
513C42—1221no—
514C42—2441no—
515C42—2441no—
516C42C23×D6111yescharacteristic
517C23—21no—
518C23—41no—
519C23—41no—
520Q8—21no—
521Q8—21no—
522Q8—1221no—
523Q8—1221no—
524Q8—9761no—
525Q8—4881no—
526Q8—2441no—
527Q8—2441no—
528Q8C2×#1175820012yes—
529Q8C2×#1175820012yes—
530D4—21no—
531D4—21no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—82no—
541D4—82no—
542D4—81no—
543D4—81no—
544D4—161no—
545D4—161no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4C22×C22⋊D6111yescharacteristic
561C2×C4#1451982211yescharacteristic
562C8—21no—
563C8—41no—
564C8—1221no—
565C8—1221no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C8—2441no—
570C8C2×#1175820012yes—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22D4×C22⋊D6111yescharacteristic
581C4—22no—
582C4—22no—
583C4—21no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—4881no—
588C4—4881no—
589C4—2441no—
590C4—2441no—
591C4C22×#1175820011yescharacteristic
592C4#1684055911yescharacteristic
593C4#1684056911yescharacteristic
594C4#1684057311yescharacteristic
595C2—81no—
596C2—161no—
597C2—161no—
598C2#2145740411yescharacteristic
599C2D4×#1175820011yescharacteristic
600C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3287C2
socleC22×C61D42
hypercenter#3287D61
hypocenterC61#19770
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3287

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21458800C2semidirect
2. #21458749C2semidirect
3. #21458486C2semidirect
4. #21445445C2semidirect
5. #21445300C2semidirect
6. #21445233C2semidirect
7. #21445194C2semidirect
8. #21445165C2semidirect
9. #21445019C2semidirect
10. #21444946C2semidirect
11. #21444914C2semidirect
12. #21444896C2semidirect
13. #21444655C2semidirect
14. #21444638C2semidirect
169. Q16×C61SD16semidirect
367. C61#19770semidirect
470. Q16#11758200semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214588002
#214587492
#214584862
#214454452
#214453002
#214452332
#214451942
#214451652
#214450192
#214449462
#214449142
#214448962
#214446552
#214446382
C61×#32872
#1977061